Tree Gaps and Orchard Problems - Numberphile

Подписаться 4,6 млн

Просмотров 828 тыс.

50% 1

Catch a more in-depth interview with Ben Sparks on our Numberphile Podcast: • The Happy Twin (with B...
Get 10% off at Squarespace: www.squarespac...
More links & stuff in full description below ↓↓↓
Ben Sparks Twitter:
/ sparksmaths
More of Ben on Numberphile:
• The Feigenbaum Constan...
• Chaos Game - Numberphile
Geobra file relevant to this video: www.geogebra.o...
Infinity with James Grime: • Infinity is bigger tha...
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumb...
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
NUMBERPHILE
Website: www.numberphile...
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberph...
Videos by Brady Haran
Patreon: / numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanb...
Sign up for (occasional) emails: eepurl.com/YdjL9

Опубликовано:

21 сен 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 1,3 тыс.

@HasekuraIsuna 6 лет назад

Pi, fibonacci, golden ratio, probability, magnitudes of infinity, Riemann zeta function... it's like all these years of watching numberphile has prepared us for this one video lol

@Ulkomaalainen 5 лет назад

I was expecting the probability of 1/e though, Euler's number is sadly missing.

@HuskyNET 5 лет назад

HasekuraIsuna 😄

@thatoneguy9582 5 лет назад

HasekuraIsuna *everyone is here*

@clockworkkirlia7475 4 года назад

@@Ulkomaalainen Pi is here and so is 0, so we just need to imagine really hard and... oh, there it is!

@Triantalex 11 месяцев назад

false.

@sethgrasse9082 6 лет назад

This infinite orchard almost solved world hunger, but unfortunately the harvesters couldn't find any trees since they were all points and had a 0% chance of being seen.

@furrane 6 лет назад

Yeah, no.

@joxfon 6 лет назад

The tree must be growing in a logarithmic scale. If they expected an infinitesimal amount of time they may only see trees on the field.

@medexamtoolscom 5 лет назад

But the fruit were all poisonous anyway.

@corpsiecorpsie_the_original 5 лет назад

How does a person pick a fruit off a point tree? I'm glad you ask. Here's another case where we want pie but pi shows up. Here's the proof....

@infinitesimotel 5 лет назад

Dismantle the food industry and throw your TV out the window, that is the only way to solve "world hunger".

@adamweishaupt3733 6 лет назад

If a tree falls in an infinite forest but you're looking in an irrational direction, does it make any sense?

@_PsychoFish_ 6 лет назад

You, Sir, just made my day xD

@mikeguitar9769 6 лет назад

Applied math has application, but pure math is completely useless. :)

@waterlubber 6 лет назад

but fun!

@vorpal22 5 лет назад

@@mikeguitar9769 It actually isn't; it's just that the uses of it often come 100 - 300 years after the math itself is discovered. For example, abstract algebra is pure math, and it's used all over cosmology and fundamental physics, e.g. to identify particles in particle collisions.

@alansmithee419 4 года назад

Your question is irrational

@tibees 6 лет назад

Somewhat unrelated but I was told by a guy who works in forestry that sometimes trees are planted in a fibonacci arrangement to maximise sunlight exposure. In a spiral like that seen in the centre of a sunflower

@slinkytreekreeper 6 лет назад

Spiral yes but single Fibonacci spiral would get too wide to be efficient really quickly leaving big spaces. The only way it could work is multiple interlaced sprials like Roger Penrose examples. Otherwise rows and columns is always more efficient which is why no commercial places use other methods unless it's stacked rows and columns.

@ChrisTian-uw9tq 6 лет назад

more efficient in the respect of harvesting and tending to the crop I guess... getting machinery/equipment around a spiral compared to up and down in rows :)

@TheAnantaSesa 6 лет назад

If the sun stayed still. But the relative motion makes any "most efficient" arrangement only temporary until a different epicenter would need selected to maximize light gain.

@MattMcConaha 6 лет назад

But surely there is an arrangement (or set of arrangements) which are on average most efficient.

@TheAnantaSesa 6 лет назад

+Matt McConaha; yeah, by alternating the rows w rows that are offset by half a tree's width.

@kevinpotts123 6 лет назад

I love the "mindfuck" aspect of mathematics and I always have. It's stuff like this where reality and intuition are on complete opposite ends of the spectrum that I love the most.

@vocalcords7397 6 лет назад

I know words, I have the best words. Nobody respects women more than me. I am the least racist person who you have ever met. Nobody lies better than me. Believe me. Sad!

@mikeguitar9769 6 лет назад

>where reality and intuition are on complete opposite ends of the spectrum Funny, that's also the feeling you get when you find an inconsistency. The moment when sh*t blows up because it's a logical fallacy.

@TheAnantaSesa 6 лет назад

In this case the example is on the other side of reality since nowhere in the real world are there point width trees to make this example even realistic. Our intuition is right for realistic examples. But for theory like the e.g. then intuition might not get us to the right answer.

@heathrowspottersam9074 6 лет назад

Kevin Potts i

@TheAnantaSesa 6 лет назад

+Dole Pole; i would call anything experienced or imagined reality but there is physical reality that has tactile reification whereas abstract theories dont. We can experience an integer in our mind (or imagination land in southpark).

@fprintf 6 лет назад

This was brilliantly presented and really fun. I would never think of this type of problem but I am super glad to have stumbled upon the fact that this kind of thinking exists!

@vocalcords7397 6 лет назад

I know words, I have the best words. Nobody respects women more than me. I am the least racist person who you have ever met. Nobody lies better than me. Believe me. Sad!

@DaveCurran 6 лет назад

Here is Douglas Adams using the same maths: “It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.”

@pragha1 6 лет назад

This doesn't sound right. Any percentage of infinity is infinity. Therefore, the number of planets that are inhabited is not finite, if the universe is infinite.

@DaveCurran 6 лет назад

Please address all complains to Douglas Adams.

@pragha1 6 лет назад

Sure. But, I didn't know he was so poor in maths. :-)

@lunafoxfire 6 лет назад

I disagree with his assertion that not every world being inhabited implies that a finite number of worlds are inhabited. I MEAN UH, COMEDY.

@danpowell806 6 лет назад

If there are an infinite number of inhabited worlds, there are an infinite number of beings. But in your lifetime you will only meet a finite number of beings. The chances of you meeting a given being are therefore zero, and it follows that any person you think that you've met is the product of a deranged imagination.

@koyouko 6 лет назад

The least close line being the golden ratio... wow

@eggory 6 лет назад

What does it mean that the golden ratio is "the least well approximated by a rational number"? I'd like to see a video just about that. It sounds like a very interesting property.

@DDranks 6 лет назад

It is indeed! The youtuber +Mathologer has done a video about this.

@amawalpe 6 лет назад

If there is a « least well approximated by a rational number » , is there a « best well approximated by a rational number » ??

@alexanderf8451 6 лет назад

The rational number p/q is a best rational approximation of some real number x if it is closer to x than any other rational number with a smaller denominator (as the denominator gets larger you can get more precise). The golden ration is the least well approximated in the sense that the best rational approximations are the worst possible. Some numbers converge as slowly but none more slowly.

@alexanderf8451 6 лет назад

There are countably infinite numbers that are best approximated by rational numbers. We call them the rational numbers!

@threepointonefour607 6 лет назад

Alexander F what about 2.618 etc ? Aka 1 + phi. Does this have the similar propert?

@pierrestober3423 6 лет назад

this was surprisingly interesting, well done. For those wondering why the golden ration is the "most irrational", it's because of its continued fraction. The golden ratio can be expressed as such: phi=1+1/(1+1/(1+1/1+1/(1+1/(............ extending to infinity. If you stop somewhere (say after n steps and ignore the rest), you get a rational approximation of the golden ratio. The fact is that the smaller the numbers you have in the continued fraction, the worse the approximation. Because there are only ones, this is the most irrational number. Hope I made myself clear.

@ardenthebibliophile 6 лет назад

I think if you’re allergic to apple trees and find yourself in the middle of an infinite apple forest you’ve made some wrong choices in life!

@corpsiecorpsie_the_original 5 лет назад

Or the DMT has kicked-in not hard enough

@AXEUROLder 5 лет назад

[Record scratch] [Freeze frame] See that guy right there? Looks like he made some wrong choices in life. Well that guy is actually me.

@moonlightcocktail 3 года назад

Doctor King?

@happygimp0 3 года назад

At least you don't see a single tree if you do not look at one on purpose. Maybe we are all in a infinite apple tree forest but we never notice it since no one ever saw a tree?

@mheermance 6 лет назад

Every direction you look you won't see a tree sounds like something out of the Hitchhiker's guide.

@mishaptrap646 6 лет назад

"There is unrest in the forest, there is trouble with the trees, for the maples want more sunlight, and the oaks ignore their pleas."

@BipinRoshan 6 лет назад

I am amazed how a simple problem had so many underlying principles involved. One of the best videos on Numberphile.

@benjaminramsey4695 6 лет назад

Totally mindbending, fantastic, wonderful! Also, I feel like an idiot - until today (age 41), I thought pi = 22/7, then watched this video, paused it halfway through, did some Googling, and mind blown, 22/7 is only a lucky approximation! But I thought that was how you actually calculated pi, just do the math for 22/7 for many many decimal places. Nope. WRONG. Calculating pi is actually MUCH harder than that, something no one ever taught me in school or in the years inbetween. So just that much more love for this particular video for opening my eyes!

@thegentleone8801 6 лет назад

Cast: Riemann's Zetafunction, the Golden Ratio, the Fibonacci Sequence, Pi, Primes

@sebastianespejoloyaga7603 6 лет назад

So many shout outs to Dr. James Grime. It's like he knows he's the best Numberphiler.

@sofarky 6 лет назад

He and Matt are the best

@julian_ossuna 6 лет назад

What about Matt Parker?

@sebastianespejoloyaga7603 6 лет назад

Future Astronaut He is a Parker Square of a Numberphiler, he's almost the best, but not quite.

@rotcod2886 6 лет назад

I think you'd just say he's the best Numberphile.

@lewismassie 6 лет назад

That was pretty cool actually. Weird maths turning up in places you don't expect is always great fun

@Alramech 6 лет назад

Awesome video. I feel like this is the math equivalent of a crossover episode. A lot of our favorite recurring characters are back: Reimann zeta function, pi, golden ratio, Fibonacci sequence....

@manuelbrand972 6 лет назад

Moral of the story: Don't plant infinitely many infinitely thin trees in a square pattern, or there will be a huge number of people walking into them, because all they see is gaps and they don't know the exact gradient they have to walk or just miss it. Invasion of the invisible trees incoming...

@koolguy728 6 лет назад

and because they're infinitely thin, the strain applied upon walking into one will be infinite, slicing anyone who would be so unfortunate clean in half.

@namewarvergeben 6 лет назад

But because they're infinitly thin, their stiffness in bending is infinitly small, so might just not feel anything after all!

@danpowell806 6 лет назад

Fortunately the people sliced in half by an infinitely thin tree get better, since zero cells were harmed by the slice.

@manuelbrand972 6 лет назад

@Dan Powell Actually you're right... So you can walk through something without even realizing it? That would be hillarious! But I think it comes down to the question of "what's the smallest thing that makes everything up?" or "what are the smallest things that make everything up?". And with that: What happens if you cut those things in half? If it would cause something like a error in the matrix of the universe it could get really ugly... But I think you only would have two halfs noone cares about :D

@danpowell806 6 лет назад

If there is a smallest thing that makes everything up, what does it mean to cut it in half?

@TyTheRegularMan 6 лет назад

I can't get enough of this guy.

@Robotomy101 4 года назад

me neither

@reabkire 6 лет назад

the intertextuality these problems display blows my mind. from irrational numbers to riemann to the golden ratio.

@Bartooc 6 лет назад

And Fibonacci Sequence.

@PC_Simo Год назад

@@Bartooc Which is pretty closely linked to the golden ratio. Not that mindblowing, if you ask me.

@DailyDrumLesson 6 лет назад

There is a German saying: "Den Wald for lauter Bäumen nicht sehen" / "To not see the forest because of too many trees" .. This suddenly makes sense.

@FrogsOfTheSea 6 лет назад

Daily Drum Lesson that’s also a saying in English - usually phrased as “can’t see the forest for the trees”

@DailyDrumLesson 6 лет назад

Didn't know that, thanks for the info!

@wouterdeniz 6 лет назад

Or in Dutch: ik zie door de bomen het bos niet meer - I don’t see the trees through the forest.

@Risu0chan 6 лет назад

"C'est l'arbre qui cache la forêt", French expression; litterally "this is the tree that hides the forest"

@bvlampe6801 6 лет назад

vor*

@ShotterManable 6 лет назад

Can't be more grateful for your videos. You made me feel like a child again with this very well edited videos. You guys, make RU-vid great!! Thanks a lot for contribute to this little nerd community

@theCodyReeder 6 лет назад

So if I'm understanding this correctly you would see no trees since in-order to see something that is a point (infinitely thin trunk) you would need to have them in every direction you look so every point is blocked out and you see a "solid" wall but since there are infinitely more gaps due to irrational fractions than there are blocked points you see no trees. That is wild isn't it!?

@waterlubber 6 лет назад

I think that any video that has an appearance of the Riemann Zeta function, phi, or "least rational numbers" is guaranteed to have those weird relationships. It's one of the more exciting areas of math, honestly.

@IDNeon357 6 лет назад

I think it's better to think of it like an atmosphere of trees....you see a similar effect with a large number of atoms occupying far less space than is otherwise empty. Yet we still see the atmosphere emerge. Granted tho. Even an atom is infinitely bigger than a point

@SilverLining1 6 лет назад

Sorry, but you've misunderstood the video. You can see any tree... if you look directly at them. Problem is, we didn't say were looking directly at them and, in fact, were directly looking at random points. Think about it like this: take a step a meter forward. What is the chance that you walked exactly one meter? Yes, it's possible, but it's completely unrealistic to imagine that ever happening because it's a single point and we have no way to connect our steps with exact distances. When you look in a random direction, you're mimicking that inability to precisely pick a point, though it is still always possible.

@thomasi.4981 6 лет назад

@@SilverLining1 You bring up an important question: can any event with an infinitesimal chance ever succeed? It is more than 0 by definition but 1 over infinity is ridiculously small. I don't think such a thing is actually possible, even in a universe where infinity makes sense. As was said in the video, the chance of seeing a tree is 0%. So there seems to be no misinterpretation.

@IDNeon357 5 лет назад

The chance of seeing a tree is 0%. But humans have a CHOICE. We can choose to look at the tree regardless what the probability is. And that is why humans have such a problem understanding probability. Because the universe isnt random but it is chosen and ordered according to intelligence.

@smoorej 5 лет назад

So let me see if I have this... he has managed to clearly and understandably present, in 14 minutes, a topic that includes geometry, trigonometry, orders of infinity, pi, the golden ratio, the Fibonacci sequence, and last but certainly not least, the Riemann Zeta function. Absolutely brilliant.

@PiTdeLyX 6 лет назад

In german we say "Man sieht den Wald vor lauter Bäumen nicht mehr" which roughly translated means "you can't see the forrest through/because of all those trees" - finally i get something that validates that saying xd

@zedbody 6 лет назад

We have the same in english- "Can't see the forest for the trees"

@devilmonkey471 6 лет назад

This might be one of the best videos I've seen in terms of tiny mind-blowing factoids.

@AlanKey86 6 лет назад

I'm going to look along a gradient of TREE(3) Please be patient whilst I just calculate how to angle myself..

@felicitas206 6 лет назад

!remindme AA(187196) seconds

@Booskop. 6 лет назад

Hey, it's Alan Key from that pi video years ago! How are you doing?

@AlanKey86 6 лет назад

Booskop - Hello! I am alive and well!

@NoriMori1992 5 лет назад

RIP

@coffeecup1196 5 лет назад

Simple, just look at the tree on the point (1, TREE(3)), and start walking.

@shocklab 6 лет назад

Presumably the breakdown in intuition here is because when we look in a direction, we actually look in a spread of directions. We would have to be able to look in an infinitely thin line for this to make sense to us.

@bens4446 6 лет назад

Many times when traveling by train in Northern California have I watched the hypnotic patterns of passing vineyard rows and thought, "now there's some fascinating mathematics waiting to be written up". But I also suspected somebody had to have already explored this matter in detail. Thanks for pointing it out. It's one of those things that's hard to google.

@TymoteuszCzech 6 лет назад

Clickbait title: "Mathematician proves that you can't see forest for the trees" :D

@Robotomy101 4 года назад

@Anonarchist 6 лет назад

the mathematical significance of being T H I C C

@HaLo2FrEeEk 6 лет назад

This absolutely blew my mind. Like, after the golden ratio and fibonacci sequence came into play I literally had to pause and set my head on my desk for a second to gather the bits of my exploded brain back together. Well done, maths.

@chrisstar969 6 лет назад

If a mathematician walks into an orchard On a trajectory based on the golden ratio How is he going to pick apples for his π?

@alexanderf8451 6 лет назад

His arms have a length which is rational. That way he will eventually be close enough to pick some apples.

@sonaruo 6 лет назад

use his hands :P

@danpowell806 6 лет назад

His arms don't have to have a rational length, they just have to be longer than some epsilon.

@KnakuanaRka 6 лет назад

He can get as close to the nearest tree as he wants if he walks far enough, so eventually he can reach out to pick the apples.

@wospy1091 5 лет назад

You picked the low hanging fruit with that pun.

@realtenfour 6 лет назад

This video started out cool, then the stuff about the golden ratio, Fibonacci, and zeta function were mind blowing.

@misium 6 лет назад

The Fibonacci transition - mind blown!

@miaomiao5462 5 лет назад

It is 10 mins away from 3 am while I watched this video and right as he was about to mention the golden ratio I figured that is what he would say, my toes clenched with excitement and I clapped with joy. I love patterns. I miss Math classes and hope to keep growing in my understanding of Math 💗

@blobberooni 6 лет назад

Sweet. I really enjoyed this one, reminds me of some of the older numberphile videos

@aidanallen1976 6 лет назад

The number of in-jokes between different mathematical sitautions and equations is astounding. It's like they all give each other cameos.

@MattVoda 6 лет назад

Best episode yet

@brycegutierrez4677 6 лет назад

I have been watching for many years, and this might be my favorite video. I really loved this one

@nikofloros 6 лет назад

This was such a absolutely lovely video!

@XWA616G 5 лет назад

Just fantastic. When I lie in my hammock in my forest, I can always see a tree. Thank goodness.

@fullmindstorm 6 лет назад

Interesting, this makes me think of the useful irrationals we haven't discovered yet.

@DasSchorty 6 лет назад

The sudden appearence of Pi and the Golden Ratio is just beautiful!

@jahwerx 6 лет назад

very nice video which presented MANY math concepts - well done!

@OrangeC7 3 года назад

Mathematicians: "These trees are infinitely thin" Also mathematicians: "Woah you can't see any of them :o"

@SkylersRants 6 лет назад

I think it’s important to stress that not only are the trees single points, but your field of view is infinitely thin. No peripheral vision is allowed in your scenario.

@lowlize 6 лет назад

He talked about the vision as a laser beam in fact.

@SkylersRants 6 лет назад

Perhaps. I didn't notice him saying it until the end.

@wmaiwald 6 лет назад

I absolutely love this guy, he's such a clear explainer, and seems like a top bloke to have a beer with. Really down to earth.

@nymalous3428 6 лет назад

Sometimes I am astounded how many different mathematical concepts converge into each other...

@samanabutool1975 6 лет назад

Nym Alous true !!!

@samanabutool1975 6 лет назад

nym alous so true !!!

@philipkelly7369 4 года назад

of all the different mathy theory-y things that I've casually observed, this video definitely has been blowing my mind the most of all of them

@HunterJE 11 месяцев назад

I feel like the claim that the line at the golden ratio (φ) gradient "avoids" trees "the most" depends on whether your definition of "most" is exclusive or not-surely it is at least tied by 1/φ, since that's basically the same line just reflected across the diagonal

@thebeautifulgame2274 4 года назад

One of the most fun days I had as a kid was when the family was camping beside a cultivated grid of mature pine trees. We played tag there. Because of the thickness of the trees it was easy to "disappear" whenever you angled away from the person who was "it". 😋

@YourMJK 6 лет назад

Matt Parker made a video once where he approximated pi by rolling two dice and using the probability of 6/pi² for two co-primes

@masonloeffler8064 6 лет назад

matt parker and appoximating pi with increasingly insane methods make the best pait

@PC_Simo Год назад

The laser beam analogy makes more sense to me; since the human field of vision obviously has width; and thus, we’d be bound to see the trees; no matter, how thin they are (including points) 🌳.

@zooblestyx 6 лет назад

Golfers will be thrilled to hear trees are now 100% air.

@johnchancey3941 6 лет назад

Videos like this are why I love Numberphile. Taking math concepts that we are already somewhat familiar with and using them in new ways or finding them in unusual places

@cyanlastride 6 лет назад

If the golden ratio is the "least near" any trees, isn't there also another line that is equally "least near" any trees if you reflect the golden ratio along the line y=x (So the line would be first going between 1,1 and 2,1 instead of 1,1 and 1,2)? Is there a special name for that line too, like the silver ratio or something like that?

@numspacsym 6 лет назад

Wouldn't the slope of that line be simply the inverse of the golden ratio? And I don't think we need to invent another name for that number.

@hemantpandey7539 6 лет назад

Isn't that just the conjugate of the golden ratio. phi-1 or aka the magnitude of the other solution to the definition of the golden ratio

@AndersTherkelsen 6 лет назад

The golden ratio has already "captured" that solution, like Hemant mentiones: If we denote the golden ratio by φ then interestingly 1/φ = φ-1. If we ignore signs then φ is the only number with this property.

@ralphinoful 6 лет назад

Yes, (1-sqrt(5))/2, the conjugate of the golden ratio.

@chrisg3030 6 лет назад

What would be the least near line in a 3D lattice? What coordinates would the elastic lines catch on?

@Sylocat 6 лет назад

This was one of the first Numberphile videos where I actually figured out most of the answers ahead of time... albeit mostly from watching earlier Numberphile videos as well as ViHart videos.

@evenmadsen4623 6 лет назад

I can't see any trees. There's an Orchard in the way.

@ALCRAN2010 3 года назад

Can't see the trees for the orchard.

@Moohasha1 6 лет назад

I love the videos like this where numbers like pi and the golden ratio just appear out of nowhere!

@DivinePonies 6 лет назад

Seems like a Parker forrest to me. It's there, but not really.

@technodruid 3 года назад

I love the videos where they're totally unintuitive but once you hear the explaination it makes total sense.

@henami552 6 лет назад

A mathematician stands inside a forest with infinitely many trees and has no chance of seeing any of them. Brilliant!

@Sweetyfragolina 6 лет назад

This may be my favorite numberphile video

@oranj.h 6 лет назад

Me too. Covered so many number problems in one go. I didn't expect that at the start!

@fauinomata 6 лет назад

The only issue that I have with this problem (the first one), is that the question then turns into "Can you see something that is truly one-dimensional (or two-dimensional)?”, and the answer is obviously "no”.

@duffman18 6 лет назад

Fausto Inomata it's not literally meant to be about whether you can see trees. It's an analogy to demonstrate a mathematical concept.

@hectorh.micheos.1717 6 лет назад

This is the best Numberphile video, IMHO: builds and references other videos that make the topic more enjoyable if you have seen those or plants curiosity on them, is interesting on its on right and easy to grasp although without subtracting complexity to the topic. And i like Ben's subdued but ever present enthusiasm. Even though "positive" answers to a video sometimes are just "background noise" to call them something, I was really excited to see such encapsulation of the Numberphile experience.

@Gyroglle 6 лет назад

ViHart fans know that the problem is mostly "how are you going to give me a random real number?" If there's infinite digit positions, how likely is it that for a random number in R, suddenly the digit sequence becomes all 0's forever?

@OlafDoschke 6 лет назад

Yes, that's where the mathematical reality fails on the infinite nature of the decimal (and also binary) representation of irrational numbers. With limited RAM, even if you don't limit yourself to any small number of bits for a float number construct you always work with rational numbers.

@joealias2594 6 лет назад

It's funny, in school, I was a "pure math" major. When conceiving of a plane with infinitely-small points, I feel I have an easy intuition for how it works. But, start introducing "realistic" elements like the thickness of your line or sight, or thickness of the tree and oop now i have no idea. I'm glad there are practical-minded people out there because if everyone was like me humanity would be screwed.

@mikeguitar9769 6 лет назад

The way to get a continuum of points on a unit interval is to have n points each with width 1/n . Then let n go to infinity. No matter what "n" is, n*(1/n) = 1 , even inside a limit, since limits are linear.

@murugend 6 лет назад

Does this sort of explain why crystals, which are organized in lattices, are see-through?

@jakobwakob1044 6 лет назад

Ben Sparks is awesome to listen to! Please get him in another video :)

@marknic 6 лет назад

Mind blown.

@smoorej 5 лет назад

Absolutely brilliantly presented. If this person ever decided to teach math in high school he’d produce an entire new generation of mathematicians

@ItsEverythingElse 6 лет назад

The golden ratio strikes again.

@alexdog6878 6 лет назад

i love seeing twisty puzzles in the background of numberphile scenes because my desk is littered with them and it's nice to see most of these guys have similar interests

@ijuldzulfadli903 6 лет назад

I used to think of it as a child and promised to figure a solution out. What a shame somebody had already done

@ApplicationBot 6 лет назад

Most things you think about and try to figure out have already been discussed or solved. I kinda find it cool when I see a problem or a question I've been asking myself be already out there

@Yora21 5 лет назад

It always makes me happy when I see a problem where at every step my intuitive answer turns out right. Happens very rarely, so it's nice when it does. :D

@santiagoarce5672 6 лет назад

Intuitively, the only slope at which you can't see trees is at an irrational slope.

@niabride7636 6 лет назад

Hey there! Here is one perfectly average person here, repeatedly awed and entertained by math videos. Thanks for the great work you all are putting into!

@louisng114 6 лет назад

I wonder what happens if the trees' radii are not constant, but instead r(x,y), some function depending on the tree's coordinate.

@luigivercotti6410 2 года назад

Well, if the radii either shrink, or grow no more than linearly by distance, then the problem is essentially the same

@mehulbhatt7850 6 лет назад

Wow! You guys are amazing. Makes me humble to see how things are connected at fundamental level.

@GoranNewsum 6 лет назад

If an infinitly thin tree falls in the forest and you're standing at the edge, does it make a sound, and do you see it?

@alansmithee419 4 года назад

It cuts through the floor.

@marks2749 4 года назад

My friend Russel said " Yes but Only when its windy ."

@theaddies 6 лет назад

The title didn't intrigue me, but having watched the video I was immensely impressed. Very well done.

@MisterAppleEsq 6 лет назад

An infinite orchard means infinite apples, which sounds cool to me.

@vytah 6 лет назад

But they're infinitely small.

@curbynet 6 лет назад

Don't tell Princess Peach!

@K-o-R 6 лет назад

But I don't like apples.

@MisterAppleEsq 6 лет назад

+K.o.R How? Have you ever eaten one?

@K-o-R 6 лет назад

Yes. I'm more of a orange person.

@tonelemoan 5 лет назад

Fascinating as usual. Glad you mentioned thickness of the 'laser' as our field of view is much more complicated than a single thin line and of course we catch more than a single photon in one fix AND light is bent by air and even the trees themselves to some degree. And we have two eyes both with wide fields of view. In which case of course it's trees from every angle.

@yashwanthreddy5306 6 лет назад

line with gradient *2/(1+sqrt(5))* is also the farthest...

@rospice 6 лет назад

Cheers for recognizing the reflective symmetry on either side of the 45 degree angle!

@coopergates9680 5 лет назад

Any number with the same non-integer part as phi will have said property, including 1 / phi and phi^2.

@henrikwannheden7114 6 лет назад

This video is chock full of mathematical concepts.. amazing!It seems that it could be a starting point for maths altogether.

@KafshakTashtak 6 лет назад

If you're allergic to those trees and wanna dodge them, just go between two rows, and walk along them, you'll have most distance from the trees.

@thomasr5302 6 лет назад

All this infinity stuff blows my mind especially that when the trees are points you see out but when they have any thickness whatsoever you can’t see anything. I feel like I understand why Plato loves maths - it’s this tiny world of perfection that we can only imagine

@daddydewitt1920 6 лет назад

10/10 ad transition

@littlezimty 6 лет назад

This has my vote for best Numberphile video. Surprising, exciting, intuitive results, and who doesn't love a guest appearance from pi!

@tgwnn 6 лет назад

I don't think the way he describes "you will not see a tree" is intuitive. In fact, you'd see an infinite number of trees as long as your field of vision is wider than 0 (i.e., between two real numbers instead of at an exact real number).

@filipsperl 6 лет назад

If you wanted to stay away as far as possible from all trees just walk 0,5 units up and then go right lol

@robertcrist6059 2 года назад

There is a board game called Photosynthesis which is liked by most and you plant seeds, grow your trees, try to gain sunlight, and chop trees down for points. Give it a go.

@igNights77 6 лет назад

I want James back, but this guy is pretty good too.

@timhuff 6 лет назад

What's all this talk about James Grime? He happens to not be in this video but people seem pretty fixated on that. Did I miss something?

@alcesmir 6 лет назад

Interesting timing. A problem regarding this occured in a math programming contest just yesterday.

@mykasikna 6 лет назад

Alcesmire interesting.. where it happened?

@kennethsizer6217 6 лет назад

If a man is standing in an orchard minding his own business, does he still run into riemann zeta, fibonacci, pi, and the golden ratio?

@JA-kg8wo 6 лет назад

I really enjoy the way that the guys explain the concepts. Easy to follow

@TheGaboom 6 лет назад

How do we know that the golden ratio is the least well approxated by ratios?

@jburtson 6 лет назад

Yeah I feel like it’s just the best example we’re aware of

@kered13 6 лет назад

It can be proven that the golden ratio is the most difficult number to approximate by rational numbers (meaning it takes large denominators to approximate the golden ratio well). This is related to Euclid's algorithm for computing the greatest common denominator.

@Octopossible 5 лет назад

Watch numberphile's video "The Golden Ratio (why is it so irrational)"

@A.J.456 6 лет назад

I love the fact that the brown paper from the Graham's number video got framed and is hanging on the wall! Would be great to have a photocopy of that on a poster!